Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism

TL;DR

The paper delivers a direct, first-principles proof of the 1-hook scaling property for colored Alexander polynomials using the Reshetikhin-Turaev formalism. By focusing on the one-hook sector of the Young graph, it fixes all $R$-matrix blocks through Yang–Baxter relations and derives recursive, level-dependent expressions for the $2\times2$ blocks, while showing the topological-locus Schur ratio collapses for $A=1$ to a simple form. The main result, $\mathcal{A}_{\lambda}(q)=\mathcal{A}_{\Box}(q^{|\lambda|})$ for one-hook $\lambda$, follows from the universality of the $|\lambda|$-dependent substitution and the finiteness constraints at $A=1$, providing a rigorous basis for observed scaling symmetries in colored Alexander polynomials. This work strengthens connections between RT representation theory, Racah–Wigner matrices, and knot invariants, with potential implications for broader scaling properties in colored HOMFLY-PT and related quantum-group invariants.

Abstract

We prove that normalized colored Alexander polynomial (the $A \rightarrow 1$ limit of colored HOMFLY-PT polynomial) evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution $q \rightarrow q^{|R|}$. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required R-matrices.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 4.1
• proof
• Lemma 5.1
• proof
• Lemma 7.1